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Theorem opnnei 23128
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei
StepHypRef Expression
1 0opn 22910 . . . . 5 (𝐽 ∈ Top → ∅ ∈ 𝐽)
21adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3 eleq1 2829 . . . . 5 (𝑆 = ∅ → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
43adantl 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
52, 4mpbird 257 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆𝐽)
6 rzal 4509 . . . 4 (𝑆 = ∅ → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
76adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
85, 72thd 265 . 2 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9 opnneip 23127 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝐽𝑥𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1093expia 1122 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑥𝑆𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1110ralrimiv 3145 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝐽) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1211ex 412 . . . 4 (𝐽 ∈ Top → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1312adantr 480 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14 df-ne 2941 . . . . . 6 (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15 r19.2z 4495 . . . . . . 7 ((𝑆 ≠ ∅ ∧ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1615ex 412 . . . . . 6 (𝑆 ≠ ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1714, 16sylbir 235 . . . . 5 𝑆 = ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18 eqid 2737 . . . . . . . 8 𝐽 = 𝐽
1918neii1 23114 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 𝐽)
2019ex 412 . . . . . 6 (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2120rexlimdvw 3160 . . . . 5 (𝐽 ∈ Top → (∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2217, 21sylan9r 508 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2318ntrss2 23065 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
2423adantr 480 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25 vex 3484 . . . . . . . . . . . . 13 𝑥 ∈ V
2625snss 4785 . . . . . . . . . . . 12 (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
2726ralbii 3093 . . . . . . . . . . 11 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28 dfss3 3972 . . . . . . . . . . . . 13 (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
2928biimpri 228 . . . . . . . . . . . 12 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3029adantl 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3127, 30sylan2br 595 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3224, 31eqssd 4001 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
3332ex 412 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
3425snss 4785 . . . . . . . . . . . 12 (𝑥𝑆 ↔ {𝑥} ⊆ 𝑆)
35 sstr2 3990 . . . . . . . . . . . . . 14 ({𝑥} ⊆ 𝑆 → (𝑆 𝐽 → {𝑥} ⊆ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3736adantl 481 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3834, 37biimtrid 242 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑥𝑆 → {𝑥} ⊆ 𝐽))
3938imp 406 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → {𝑥} ⊆ 𝐽)
4018neiint 23112 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝐽𝑆 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41403com23 1127 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42413expa 1119 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4339, 42syldan 591 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4443ralbidva 3176 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4518isopn3 23074 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
4633, 44, 453imtr4d 294 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
4746ex 412 . . . . . 6 (𝐽 ∈ Top → (𝑆 𝐽 → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
4948adantr 480 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
5113, 50impbid 212 . 2 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
528, 51pm2.61dan 813 1 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333  {csn 4626   cuni 4907  cfv 6561  Topctop 22899  intcnt 23025  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-ntr 23028  df-nei 23106
This theorem is referenced by:  neiptopreu  23141  flimcf  23990
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